Consider a linear vector space of functions over the complex numbers. Any linear combination of functions Σaifi(x) with complex coefficients ai is also a member of the space. The functions are to be integrable, with ∫f*(x)f(x)dx < ∞, where the limits on x are ±∞. We may extend this to more than one argument x in the obvious way, but we keep to one argument for simplicity. An inner product of any two functions (a,b) = ∫f*afbdx is defined. This space is, therefore, a Hilbert space, with many similarities with an ordinary Euclidean vector space. The norm of a function is ||f(x)|| = (f,f) (or its square root). We can multiply any function by a constant to make its norm unity.

The space may be spanned by a properly chosen infinite set of basis functions {ui} that satisfy (ui,uj*) = δij and the completeness relation δ(x - x') = Σui(x)ui(x'). This allows us to expand any function as f(x) = ∫f(x')δ(x - x')dx' = Σui(x)∫ui*(x')f(x')dx'= Σaiui(x), where the amplitudes ai = (ui,f) are analogous to a vector.

Let A be an operator that converts any function f(x) into another function g(x) that is also in the space, g(x) = Af(x). Since we can express f(x) as a linear combination of the basis functions ui(x), we can predict the result of operating with A if we know the effect of A on the basis functions. This can be expressed by expanding Aui in terms of the ui, obtaining the amplitudes (ui,Auj, which form a matrix usually written (i|A|j) = ∫ui*(x)Auj(x)dx.

Note that the operator A operates on the function to the right. To find an expression giving the same value in which A operates on the function to the left, we define the adjoint operator A by (i|A|j) = ∫[Aui(x)]ujdx. Taking the complex conjugate, we have (i|A|j)* = ∫ujAui(x)dx =(j|A|i). This shows that the matrix of A is the transposed complex conjugate of the matrix of A. It follows from this that if A is nonsingular so A-1 exists, so does A†-1.

Since (Ai,j) = (i,Aj) = (A††i,j), we have A†† = A. Also, (i,ABj) = (Ai,Bj) = (BAi,j), or (AB)= BA. Applying this to AA-1 = 1, we find that 1 = (AA-1) = A-1A, or A-1† = A†-1. Furthermore, (i,λA) = (λ*Ai,j) from the definition. Finally, (i,[A + B]j) = ([A + B]i,j), or (A + B) = + B. These are the most important algebraic properties of the adjoint operator.

If A = A, then (i|A|j)* = (j|A|i), so the matrix of A is Hermitian, and A is called a Hermitian operator. The diagonal matrix elements are clearly real, (i|A|i)* = (i|A|i), and the eigernvalues of the matrix are also real. This is the reason Hermitian matrices are so useful in quantum mechanics, where they correspond to real dynamical variables.

For example, consider the operator i(d/dx). Its matrix elements are ∫u*i(dv/dx)dx, where we write u,v for ui and uj. Since d(u*v)/dx = u*(dv/dx) + v(du*/dx), the integral us equal to ∫[d(u*v)/dx-iv(du*/dx)]dx = ∫-iv(du*/dx)dx, as the integrated part vanishes at x = ±∞. This may be written ∫(idu/dx)*v dx, so the adjoint operator is also i(d/dx) and the operator is Hermitian. When multiplied by -h/2π, this is the operator for the linear momentum corresponding to the coordiante x. The presence of the i in the operator compensates for the change of sign on the integration by parts that throws the differentiation onto the first function. On the other hand, the coordinate operator x is clearly Hermitian since it moves from one function to the other without change. The second derivative operator d2/dx2 is Hermitian, since two integrations by parts do not change the sign. Therefore, the Laplacian, del2, is also Hermitian. It follows that the single-particle Hamiltonian H = (h/2π)2(del2/2m) + V(r) is Hermitian.

It is easy to show that the components of the angular momentum operator L = r x p are Hermitian. For example, Lz = xpy - ypx so Lz = pyx - pxy = xpy - ypx = Lz, since the coordinates and momentum components commute, and are themselves Hermitian.

If an operator A is not Hermitian, the combination (A + A)/2 will be Hermitian. The adjoint of an operator is analogous to the complex conjugate of a number, and an operator can be resolved into Hermitan and anti-Hermitian parts analogous to real and imaginary parts of a complex number. An operator and its adjoint are evidently quite similar to each other and much like a complex conjugate.

An interesting example is provided by the Runge-Lenz vector, a constant of the motion in orbital motion under an exact inverse-square force. This vector is classically defined as M = (p x L)/m - kr/r, where the attractive potential is k/r. The second term is clearly Hermitian. The z-component of the first term is Mz = (pxLy - pyLx)/m. Taking the adjoint and using the Hermitian nature of p and x, we find Mz = (Lypx - Lxpy)/m, or M = -(L x p)/m. Classically, this would show the operator was Hermitian, but quantum mechanically p and L do not commute because of the presence of x in L. A Hermitian M can be constructed as in the preceding paragraph of which the first term is (p x L - L x p)/2m, and this operator remains a constant of the motion.

Even though r and p do not commute in general, it happens that r x p = -p x r, as can be seen by writing out the components. Taking the adjoint interchanges the bottom two rows of the cross product determinant, so this property makes L Hermitian.

### References

L. I. Schiff, Quantum Mechanics, 3rd ed. (New York: McGraw-Hill, 1968), Chapter 6. The Runge-Lenz vector appears on p. 236.

There are Wikipedia articles on Adjoint operators and Hilbert space, and other similar topics that are worth looking at.